An Interactive Train Scheduling Tool for Solvingand Plotting Running Maps?

F. Barber1, M.A. Salido2, L. Ingolotti1, M. Abril1, A. Lova3, P. Tormos3

1 DSIC, 3 DEIOAC, Universidad Politecnica de Valencia, Spain2 DCCIA, Universidad de Alicante, Spain

{fbarber, msalido, lingolotti, mabril }@dsic.upv.es{allova, ptormos}@eio.upv.es

Abstract. We present a tool for solving and plotting train scheduleswhich has been developed in collaboration with the National Network ofSpanish Railways (RENFE). This tool transforms railway problems intoformal mathematical models that can be solved and then plots the bestpossible solution available. Due to the complexity of problems of thiskind, the use of preprocessing steps and heuristics become necessary.The results are plotted and interactively filtered by the human user.

1 Introduction

Over the last few years, railway traffic have increased considerably, which hascreated the need to optimize the use of railway infrastructures. This is, how-ever, a hard and difficult task. Thanks to developments in computer science andadvances in the fields of optimization and intelligent resource management, rail-way managers can optimize the use of available infrastructures and obtain usefulconclusions about their topology.

The overall goal of a long-term collaboration between our group at the Poly-technic University of Valencia (UPV) and the National Network of Spanish Rail-ways (RENFE) is to offer assistance to help in the planning of train scheduling,to obtain conclusions about the maximum capacity of the network, to identifybottlenecks, to determine the consequences of changes, to provide support in theresolution of incidents, to provide alternative planning and real traffic control,etc. Besides of mathematical processes, a high level of interaction with railwayexperts is required to be able to take advantage of their experience.

Different models and mathematical formulations for train scheduling havebeen created by researchers [9], [3], [4], [8], [6], [2], [5], [1], [10], etc. SeveralEuropean companies are also working on similar systems. These systems in-clude complex stations, rescheduling due to incidents, rail network capacities, etc.These are complex problems for which work in network topology and heuristic-dependent models can offer adequate solutions.

? This work has been supported by a join contract RENFE-UC/UPV and a visitingresearch fellow (Miguel A. Salido) from the Polytechnic University of Valencia, Spain.

In this paper, we describe a specific tool for solving and plotting optimizedrailway running maps. A running map contains information regarding the topol-ogy of the railways (stations, tracks, distances between stations, traffic controlfeatures, etc.) and the schedules of the trains that use this topology (arrivaland departure times of trains at each station, frequency, stops, junctions, cross-ing, overtaking, etc,)(Figure 1). An optimized running map should combine userrequirements with the existing constraints (railway infrastructures, user require-ments and rules for traffic coordination, etc.). In our system, the railway runningmap problem is formulated as a Constraint Satisfaction Problem (CSP) to beoptimized. Variables are frequencies, arrival and departure times of strains atstations. The parameters of the process are defined using user interfaces anddatabase accesses. The problem formulation is then translated into a formalmathematical model to be solved for optimality by means of mixed integer pro-gramming techniques. Due to the dimensions of the problem and the complexnature of the mathematical models, several preprocesses and heuristic criteriaare included in order to obtain good solutions in a reasonable time. The usercan also modify the obtained timetable so that the system interactively guaran-tees the fulfillment of problem constraints by detecting whether a constraint orrequirement is violated. Several reports can be obtained from the final timetable.

2 Problem Statement

A sample of a running map is shown in Figure 1, where several train crossings canbe observed. On the left side of Figure 1, the names of the stations are presentedand the vertical line represents the number of tracks between stations (one-wayor two-way). The objective of the system is to obtain a correct and optimizedrunning map taking into account: (i) the railway infrastructure topology, (ii)user requirements (parameters of trains to be scheduled), (iii) traffic rules, (iv)previously scheduled traffic on the same railway network, and (v) criteria foroptimization.

A railway network is basically composed of stations and one-way or two-waytracks. A dependency can be:

– Station: Place for trains to park, stop or pass through. Each station isassociated with a unique station identifier. There are two or more tracks ina station where crossings or overtaking can be performed.

– Halt: Place for trains to stop, pass through, but not park. Each halt isassociated with a unique halt identifier.

– Junction: Place where two different tracks fork. There is no stop time.

In Figure 1, horizontal dotted lines represent halts or junctions, while con-tinuous lines represent stations. On a rail network, the user needs to schedulethe paths of n trains going in one direction and m trains going in the oppositedirection, trains of a given type and at a desired scheduling frequency.

The type of trains to be scheduled determines the time assigned for travelbetween two locations on the path. The path selected by the user for a train trip

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determines which stations are used and the stop time required at each stationfor commercial purposes. New trains to be scheduled must be coordinated withpreviously scheduled trains. In order to perform crossing or overtaking in asection with a one-way track, one of the trains should wait in a station. This iscalled a technical stop. One of the trains is detoured from the main track so thatthe other train can cross or continue (Figure 2).

2.1 Railway Traffic Rules, topological and requirements constraints

A valid running map must satisfy and optimize the set of existing constraints inthe problem. Some of the main constraints to be considered are:

1. Traffic rules guarantee crossing and overtaking operations. The main rulesto take into account are:

– Crossing constraint : Any two trains (Ti and Tj) going in oppositedirections must not simultaneously use the same one-way track (A-B).

TiArrivesA < TjDeparturesA or TjArrivesB < TiDeparturesB

The crossing of two trains can be performed only on two-way tracks andat stations, where one of the two trains has been detoured from the maintrack (Figure 2). Several crossings are shown in Figure 1.

– Overtaking constraint : Any two trains (Ti and Tj) going at differentspeeds in the same direction can only overtake each other at stations.

TiDeparturesA < TjDeparturesA → TiArrivesB < TjArrivesB

The train being passed is detoured form the main track so that the fastertrain can pass the slower one (see Figure 2).

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– Expedition time constraint . There exists a given time to put a de-toured train back on the main track and exit from a station.

– Reception time constraint . There exists a given time to detour a trainfrom the main track so that crossing or overtaking can be performed.

– Succession time constraint . Any two trains traveling in the samedirection must respect the safety headway between trains (even if speedsare different). This should be maintained at arrival times, departurestimes and along the entire path. The succession time constrain dependson the features of traffic control in each section between two locations(manual, automatic, etc.)

2. User Requirements: The main constrains due to user requirements are:– Type and Number of trains going in each direction to be scheduled

and Travel time between locations.– Path of trains: Locations used and Stop time for commercial pur-

posed in each direction.– Scheduling frequency . The frequency requirements of the departure of

trains in both directions This constraint is very restrictive, because, whencrossing and overtaking are performed, trains must wait for a certain timeinterval at stations. This interval must be propagated to all trains goingin the same direction in order to maintain the established schedulingfrequency. The user can require a fixed frequency, a frequency within aminimum and maximum interval, or multiple frequencies.

– Time interval for the departure of the first train going in one directionand the departure of the first train going in the opposite direction.

– Maximum slack . This is the maximum time allowed to perform all thetechnical operations.

3. Topological railways infrastructure and type of trains to be scheduledgive rise other constraints to be taken into account. Some of them are:

– Number of tracks in stations (to perform technical and/or commercialoperations) and the number of tracks between two locations (one-way ortwo-way). No crossing or overtaking is allowed on a one-way track,

– Closing times in the locations, when no technical and /or commercialoperations can be performed,

– Added time for the stopping and starting process of a train due toan unexpected/unscheduled technical stop.

In accordance with user requirements, the system should obtain the bestsolutions available so that all the above constraints are satisfied. Several criteriacan exist to qualify the optimality of solutions: minimize duration and/or numberof technical stops, minimize the total time of train trips (span) of the totalschedule, giving priority to certain trains, etc.

2.2 General System Architecture

The general outline of our system is presented in figure 3. It shows several steps,some of which require the direct interaction with the human user to insert re-quirement parameters, parameterize the constraint solver for optimization, ormodify a given schedule. First of all, the user should require the parameters ofthe railway network and the train type from the central database (Figure 4).This database stores the set of locations, lines, tracks, trains, etc. Normally, thisinformation does not change, but authorized users may desire to change thisinformation. With the data acquired from the database, the system carries out apreprocessing step, in which a linear constraint solver for optimization is used tosolve a linearized problem in order to complete the formal mathematical model.In the processing step, the formal mathematical model is solved by a mixed-integer constraint solver for optimization, returning the running map data. Ifthe mathematical model is not feasible, the user must modify the most restric-tive problem parameters. If the running map is consistent, the graphic interfaceplots the scheduling (Figure 6). Afterwards, the user can graphically interactwith the scheduling to modify the arrival or departure times. Each interactionis automatically checked by the constraint checker in order to guarantee theconsistency of changes. The user can finally print out the scheduling, to obtainreports with the arrival and departure times of each train in each location, orgraphically observe the complete scheduling topology.

3 Optimization Process: Preprocesses and Heuristics

The two main issues in our problem are (i) the specification of the model accord-ing to the existing constraints, and (ii) the constraint solver for optimization,which requires mathematical techniques, criteria and heuristics. This problemis more complex than job-shop scheduling [7], [10]. Here, two trains, travelingin opposite directions, as well as two trains traveling in the same direction usetracks between two locations for different durations, and these durations are

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causally dependent on how the scheduling itself is done (ie: order of tasks), dueto the stopping, and starting time for trains in a non-required technical stop,expedition, reception, succession times, etc. Some processes (detour from themain railway) may or may not be required for each train at each location. In oursystem, the problem is modeled as a CSP, where finite domain variables repre-sent frequency and arrival and departure times of trains of locations. Relationson these variables permit the management of all the constraints due to the userrequirements, topological constraints, traffic rules, commercial stops, technicaloperation, maximum slacks, etc. Hundred of trains, of different types and speeds,in different directions, along paths of dozens of stations have to be coordinated.Thus, many variables, and many and very complex constraints arise. The prob-lem turns into a mixed-integer programming problem, in which thousands ofinequalities have to be satisfied and a high number of variables take only integervalues. As is well known, this type of model is far more difficult to solve thanlinear programming models. Depending on the problem size, some pre-processesare performed in the system in order to reduce the number of variables and thecomplexity of the complete mathematical model:

1. Topological and geometrical processes. Identification of bottlenecks, period-icity of running maps, possible wide-paths for trains, etc.

2. Linear programming approach, in order to identify potential crossing or over-taking. Important decisions in the linear problem are the probabilistic fea-sibility of stations where crossing or overtaking can be performed, optimalinitial departure times to minimize intersections, frequency, etc.

Afterwards, the optimization process is performed in several ways accordingto the level of required solutions and the problem size:

1. Complete: The process is performed taking into account the entire problem.2. Incremental: The process performs an incremental coordination of trains.3. Iterative: The solution is obtained by replicating a pattern found in a previ-

ous process of a simpler problem.

In addition, several heuristics and prioritization criteria are applied. Thesepre-processes as well as the level of heuristics applied in the optimization pro-cess can be selected by the user (Figure 5). However, the system can also au-tomatically recommend or select the appropriate choices depending on differentparameters and the complexity of the problem.

4 Functionalities and System Interfaces

In this section, we make a brief description about the functionalities and someinterfaces of the developed tool:

1. Specify a demand (Fig. 4). Through several interfaces, the user providesdata about requirements of a demand: Number and type of trains to bescheduled, path, direction, frequency, commercial stops, time interval forthe initial departure of the first train, maximum slacks, previous demandsto be considered on the same network, etc.

Fig. 4. Some user requirement interfaces

2. Parameterization of optimization process. (see Figure 5 left). The optimiza-tion process can be parameterized to bound the search space and/or theexecution time, to perform an incremental search, etc. During the solvingprocess, using mathematical optimization packages, its state is displayed(Figure 5 upper right), and when it ends, the information about its finalstate is shown. (Figure 5 lower right).

Fig. 5. Some interfaces for the optimization process

5 Evaluation

The application and performance of this system depends on several factors:Railway topology (locations, distances, tracks, etc.), number and type of trains(speeds, starting and stopping times, etc.), frequency ranges, initial departureinterval times, maximum slacks, etc. Several running maps are shown in this sec-tion. Figure 6 shows the schedule obtained from a requirement for an establishedfrequency of only one type of train, in both directions. Thus, only crossing prob-lems appear in one-way track sections and are performed in adjacent stations.Time slacks for technical operations due to crossing are minimized. The runningpath shown is optimal for the given parameters.

The system allows the user interactively modify an obtained running map (seeFigure 6 lower right). A range between any two stations can be selected, so theuser can modify the arrival and departure times between them, the commercialand/or technical stops, etc. Interactively, the system checks these changes toassure constraint fulfillment, so that non-valid running maps that are not inaccordance with traffic rules, topological and train constraints are allowed. Thisfeature permits the user to adapt running maps to special circumstances and totry alternative scheduling, etc.

Figure 7 (left) displays a running map of two different types of trains. Thus,two different requirements should be coordinated on the same rail network. Eachrequirement specifies a given number and type of train, frequency, start times,paths, location stops, etc. More specifically, a new requirement is added to therunning map in Figure 6, so that crossing and overtaking problems appear.Therefore, the system allows the user to add new requirements to a previous

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running map so that this previous running map can be modified or can be keptthe same, depending on the user’s needs. The different lines in Figure 6 and inFigure 7 correspond to the paths performed by each train, respectively, whereeach point of a line corresponds to the position of a train at a given time.

6 Conclusions

We have presented the main features of a flexible and useful tool for solving andplotting optimized train schedules in collaboration with the National Network ofSpanish Railways (RENFE). No other equivalent system is known from authors.This tool transforms railway problems into formal mathematical models. TheNP-Hard complexity of problems of this kind requires that several heuristiccriteria and pre-process be parameterized by the user or automatically selectedby the application. They are then added to the solving process in order to boundthe search space and to obtain an optimal solution.

The main features of the proposed approach are: (i) the access to databases toobtain centralized data about the rail network, trains and required paths, (ii) themodel specification for a complex problem, (iii) the various parameterizations,levels and approaches for search processes for optimization, (iv) the graphicalinterfaces and interactivity with the user. This tool, at a current stage of pre-integration, supposes the application of methodologies of Artificial Intelligencein a problem of great interest and will assist railways managers in optimizing theuse of railway infrastructures and will help them in the resolution of complexscheduling problems.

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